Suppose that $U,D,u_{*},\Lambda$ are as above and data $g_{0},J_{0},A_{0}$ has Property (H). Then there is some $\psi>0$ with the following effect. Suppose that $X,g_{X},J_{X},L,A_{X})$ is in ${\mathcal{K}}(n,c,V)$. If we can find $k>0$, an open embedding $\chi:U\rightarrow X$ and a bundle isomorphism $\hat{\chi}:\Lambda\rightarrow\chi^{*}(L^{k})$ such that

$$\|\chi^{*}(J)-J_{0}\|_{U},\|\chi^{*}(kg)-g_{0}\|_{U},\|\chi^{*}(A^{\otimes k})-A\|_{U}\leq\psi,$$

then there is a holomorphic section $s$ of $L^{k}\rightarrow X$ with $L^{2,\sharp}$ norm at most $(11/10)(2\pi)^{n}$ and with $|s(x)|\geq 1/4$ at all points $x$ a distance (in the scaled metric) less than $(4K_{1})^{-1}$ from $\chi(u_{*})$